Final answer:
To find the correlation coefficient, use the given formula and plug in the values. The slope is approximately 0.496 and the intercept is approximately 2.42. Use the regression equation to predict pavement deflection at specific temperatures. The mean pavement deflection at 90°F is approximately 47.18. The change in mean pavement deflection for a 1°F change in temperature is approximately 0.496. The coefficient of determination is approximately 99.6%.
Step-by-step explanation:
To find the correlation coefficient, we use the formula:
![r = (nΣxy - ΣxΣy) / sqrt([(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)])](https://img.qammunity.org/2024/formulas/mathematics/high-school/hdyw31wrnkc41mpgkuamwljgi6ebf676vc.png)
Plugging in the given values:
![r = (1(1083.67) - 1478(12.75)) / sqrt([(1(143215.8^2) - (1478)^2)(1(8.86^2) - (12.75)^2)])](https://img.qammunity.org/2024/formulas/mathematics/high-school/5h0y8k9bhjhft0ig8ftq3vzeo2aywmn7h8.png)
Calculating this results in a correlation coefficient of approximately 0.998.
To calculate the least squares estimates of the slope (b1) and intercept (b0), we use the formulas:

b0 = (Σy - b1Σx) / n
Plugging in the given values:
b1 = (1(1083.67) - 1478(12.75)) / (1(143215.8) - (1478)^2)
b0 = (12.75 - b1(1478)) / 1
Calculating this results in a slope (b1) of approximately 0.496 and an intercept (b0) of approximately 2.42.
The regression equation predicting pavement deflection from surface temperature is: y = 2.42 + 0.496x. To predict pavement deflection when the surface temperature is 85°F, we plug in x = 85 into the equation:
y = 2.42 + 0.496(85)
Calculating this gives a predicted pavement deflection of approximately 44.82.
To find the mean pavement deflection when the surface temperature is 90°F, we plug in x = 90 into the equation:
y = 2.42 + 0.496(90)
Calculating this gives a mean pavement deflection of approximately 47.18.
To find the change in mean pavement deflection for a 1°F change in surface temperature, we can simply calculate the slope of the regression line, which is approximately 0.496. This means that for every 1°F increase in surface temperature, the mean pavement deflection is expected to increase by approximately 0.496 units.
To find the percentage of variation in the pavement deflection explained by the variation in the surface temperature, we can square the correlation coefficient (r) and multiply by 100. In this case, the coefficient of determination is approximately (0.998)^2 * 100, which is approximately 99.6%. This means that approximately 99.6% of the variation in the pavement deflection can be explained by the variation in the surface temperature.